Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right) + \log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right) + \log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right)
double f(double x) {
        double r76498 = 1.0;
        double r76499 = x;
        double r76500 = r76498 / r76499;
        double r76501 = r76499 * r76499;
        double r76502 = r76498 - r76501;
        double r76503 = sqrt(r76502);
        double r76504 = r76503 / r76499;
        double r76505 = r76500 + r76504;
        double r76506 = log(r76505);
        return r76506;
}
↓
double f(double x) {
        double r76507 = 1.0;
        double r76508 = x;
        double r76509 = 2.0;
        double r76510 = pow(r76508, r76509);
        double r76511 = r76507 - r76510;
        double r76512 = sqrt(r76511);
        double r76513 = r76512 / r76508;
        double r76514 = r76507 / r76508;
        double r76515 = r76513 + r76514;
        double r76516 = sqrt(r76515);
        double r76517 = log(r76516);
        double r76518 = r76517 + r76517;
        return r76518;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \log \color{blue}{\left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)}\]

  4. Applied log-prod0.0

    \[\leadsto \color{blue}{\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - {x}^{2}}}{x}}\right)} + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]

  6. Simplified0.0

    \[\leadsto \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - {x}^{2}}}{x}}\right) + \color{blue}{\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - {x}^{2}}}{x}}\right)}\]

  7. Final simplification0.0

    \[\leadsto \log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right) + \log \left(\sqrt{\frac{\sqrt{1 - {x}^{2}}}{x} + \frac{1}{x}}\right)\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))